High-order Accurate Numerical Methods for Solving the Space Fractional Advection-dispersion Equation

The fractional advection-dispersion equation (FADE) is a generalization of the classical advection-dispersion equation (ADE). It provides a useful descriptionof transport dynamics in complex systems which are governed by anomalousdiffusion and nonexponential relaxation. The FADE was firstly proposed by Chaves to investigate the mechanism of super diffusion and with the goal of having a model able to generate the L´evy distribution and was later generalized by Benson et al and has since been treated by numerous authors. Many numerical methods have been proposed for solving the FADE.

Meerschaert and Tadjeran developed practical numerical methods to solve the one-dimensional space FADE with variable coefficients. Liu et al transformed the space fractional Fokker-Planck equationinto a system of ordinary differential equations (method of lines), which wasthen solved using backward differentiation formulas. Liu et al proposed an implicit difference method (IDM) and an explicit difference method (EDM) to solve a space-time FADE. Liu et al. presented a random walk model for approximating a L´evy-Feller advection-dispersion process and proposed an explicit finite difference approximation (EFDA).